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Dirac matrices
The Dirac matrices are a set of 16 matrices created from the Pauli matrices by using the . All 16 Dirac matrices square to positive one (i.e. I_4 ). \begin{array}{c|c|c|c} \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{pmatrix} & \begin{pmatrix} 0&1&0&0\\ 1&0&0&0\\ 0&0&0&1\\ 0&0&1&0 \end{pmatrix} & \begin{pmatrix} 0&-i&0&0\\ i&0&0&0\\ 0&0&0&-i\\ 0&0&i&0 \end{pmatrix} & \begin{pmatrix} 1&0&0&0\\ 0&-1&0&0\\ 0&0&1&0\\ 0&0&0&-1 \end{pmatrix} \\\hline \begin{pmatrix} 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0 \end{pmatrix} & {\color{red} \begin{pmatrix} 0&0&0&1\\ 0&0&1&0\\ 0&1&0&0\\ 1&0&0&0 \end{pmatrix} } & {\color{red} \begin{pmatrix} 0&0&0&-i\\ 0&0&i&0\\ 0&-i&0&0\\ i&0&0&0 \end{pmatrix} } & {\color{red} \begin{pmatrix} 0&0&1&0\\ 0&0&0&-1\\ 1&0&0&0\\ 0&-1&0&0 \end{pmatrix} } \\\hline {\color{red} \begin{pmatrix} 0&0&-i&0\\ 0&0&0&-i\\ i&0&0&0\\ 0&i&0&0 \end{pmatrix}} & \begin{pmatrix} 0&0&0&-i\\ 0&0&-i&0\\ 0&i&0&0\\ i&0&0&0 \end{pmatrix} & \begin{pmatrix} 0&0&0&-1\\ 0&0&1&0\\ 0&1&0&0\\ -1&0&0&0 \end{pmatrix} & \begin{pmatrix} 0&0&-i&0\\ 0&0&0&i\\ i&0&0&0\\ 0&-i&0&0 \end{pmatrix} \\\hline {\color{red} \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{pmatrix}} & \begin{pmatrix} 0&1&0&0\\ 1&0&0&0\\ 0&0&0&-1\\ 0&0&-1&0 \end{pmatrix} & \begin{pmatrix} 0&-i&0&0\\ i&0&0&0\\ 0&0&0&i\\ 0&0&-i&0 \end{pmatrix} & \begin{pmatrix} 1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&1 \end{pmatrix} \end{array} Surprisingly the 4 by 4 table above forms a multiplication table even though it is actually created by the following rules: : \begin{array}{c|c|c|c} \sigma_{00} & \sigma_{01} & \sigma_{02} & \sigma_{03} \\\hline \sigma_{10} & {\color{red} \sigma_{11}} & {\color{red} \sigma_{12}} & {\color{red} \sigma_{13}} \\\hline {\color{red} \sigma_{20}} & \sigma_{21} & \sigma_{22} & \sigma_{23} \\\hline {\color{red} \sigma_{30}} & \sigma_{31} & \sigma_{32} & \sigma_{33} \end{array} \quad \text{where} \quad \quad\quad\begin{align} \sigma_{ij} &= \sigma_i \otimes \sigma_j \end{align} where \sigma_i and \sigma_j are the original 2x2 Pauli matrices and \otimes is the (not the tensor product) : \sigma_0 = \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}, \quad \sigma_1 = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \quad \sigma_2 = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, \quad \sigma_3 = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} The Dirac matrices are commonly referred to by the following name. Note that 4 of the Dirac matrices are denoted \sigma_i even though the same symbol can refer to the original Pauli matrices. : \begin{array}{c|c|c|c} \sigma_0 & \sigma_1 & \sigma_2 & \sigma_3 \\\hline \rho_1 & {\color{red} \alpha_1} & {\color{red} \alpha_2} & {\color{red} \alpha_3} \\\hline {\color{red} \rho_2} & y_1 & y_2 & y_3 \\\hline {\color{red} \rho_3} & \delta_1 & \delta_2 & \delta_3 \end{array} \quad \text{where} \quad \begin{align} \sigma_0 &= I_4 \\ -\rho_1 &= y_5 \\ \rho_2 &= {\color{red} \alpha_5} \\ \rho_3 &= {\color{red} \alpha_4} = y_4 \end{align} The 16 original Dirac matrices form six anticommuting sets of five matrices each (Arfken 1985, p. 214). # {\color{red} \alpha_1, \alpha_2, \alpha_3,} \quad \rho_3, \rho_2 \quad ({\color{red} \alpha_4, \alpha_5}) # y_1, y_2, y_3, \quad \rho_3, -\rho_1 \quad (y_4, y_5) # \delta_1, \delta_2, \delta_3, \quad \rho_1, \rho_2 # \alpha_1, y_1, \delta_1, \quad \sigma_2, \sigma_3 # \alpha_2, y_2, \delta_2, \quad \sigma_1, \sigma_3 # \alpha_3, y_3, \delta_3, \quad \sigma_1, \sigma_2 Any of the 15 original Dirac matrices (excluding the identity matrix \sigma_0 ) anticommute with eight other original Dirac matrices and commute with the remaining eight, including itself and the identity matrix. :Source: Weisstein, Eric W. "Dirac Matrices." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiracMatrices.html Alpha multiplication table Gamma matrices The gamma matrices, \{ \gamma^0, \gamma^1, \gamma^2, \gamma^3 \} are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra C''ℓ1,3('R''').Wikipedia:Gamma matrices : \begin{align} \gamma^0 = {\color{red} \alpha_4 \alpha_0} &= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}, \quad & \gamma^1 = {\color{red} \alpha_4 \alpha_1} &= \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix} \\ \gamma^2 = {\color{red} \alpha_4 \alpha_2} &= \begin{pmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{pmatrix}, \quad & \gamma^3 = {\color{red} \alpha_4 \alpha_3} &= \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} \end{align} The first squares to one and the rest square to negative one: : (\gamma^0)^2 = I_4 \\ (\gamma^1)^2 = -I_4 \\ (\gamma^2)^2 = -I_4 \\ (\gamma^3)^2 = -I_4 For reference: : {\color{red} \alpha_4 \alpha_5} = \begin{pmatrix} 0 & 0 & -i & 0 \\ 0 & 0 & 0 & -i \\ -i & 0 & 0 & 0 \\ 0 & -i & 0 & 0 \end{pmatrix} References